Editing Pell's equation (section)

===Algebraic number theory===
Pell's equation is closely related to the theory of [[algebraic number]]s, as the formula
<math display="block">x^2 - n y^2 = (x + y\sqrt n)(x - y\sqrt n)</math>
is the [[Field norm|norm]] for the [[Ring (mathematics)|ring]] <math>\mathbb{Z}[\sqrt{n}]</math> and for the closely related [[quadratic field]] <math>\mathbb{Q}(\sqrt{n})</math>. Thus, a pair of integers <math>(x, y)</math> solves Pell's equation if and only if <math>x + y \sqrt{n}</math> is a [[Unit (ring theory)|unit]] with norm 1 in <math>\mathbb{Z}[\sqrt{n}]</math>.<ref>{{Cite journal |last=Clark |first=Pete |title=Number Theory: A Contemporary Introduction |url=http://alpha.math.uga.edu/~pete/4400FULL2018.pdf|journal=University of Georgia}} Ch. 7 The Pell Equation</ref> [[Dirichlet's unit theorem]], that all units of <math>\mathbb{Z}[\sqrt{n}]</math> can be expressed as powers of a single [[Fundamental unit (number theory)|fundamental unit]] (and multiplication by a sign), is an algebraic restatement of the fact that all solutions to the Pell's equation can be generated from the fundamental solution.<ref>{{Cite web |last=Conrad |first=Keith |title=Dirichlet's Unit Theorem |url=https://kconrad.math.uconn.edu/blurbs/gradnumthy/unittheorem.pdf |access-date=14 July 2020}}</ref> The fundamental unit can in general be found by solving a Pell-like equation but it does not always correspond directly to the fundamental solution of Pell's equation itself, because the fundamental unit may have norm −1 rather than 1 and its coefficients may be half integers rather than integers.